 That's perfect. Let me review where we left off. So last time I told you a story and the story in sort of in summary was the following that well we learn from explicit examples explicit solutions of the Einstein equations that sort of some sort of singularity happens dynamically and we learn from Penrose's incompleteness theorem that Jodhese can completeness Okay, it's not going to go away After perturbation that's to say that is a stable property. So we have to take it serious and Well, given that realization on a second look at the sort of Oppenheimer-Sneider space-time a space-time when in some sense when it was first written down it looked to be maybe pathological Actually, this is as good as it gets because quote singular behavior or Jodhese can completeness Let me say it like that is organized in a very very nice way and Sort of the cosmic censorship conjectures they conjecture that those good properties Okay, hold for generic initial data For reasonable Einstein mother systems. So what are those good properties? Well, the first good property is that despite Jodhese in completeness so there may be some Observers that only live for finite time, but if you're sufficiently far away Okay, you you you live for all time and in your past you don't see anything which is bad Okay, not in finite time so the sort of Mathematization of this statement is that there is some notion of futurnal infinity. This is the sort of Representation of far away observers in fact far away observers in the radiation zone and this is itself complete So space time may be geodesically incomplete but futurnal infinity is complete So this is true for open-heimer Snyder and weak cosmic censorship is the conjecture that this is true for generic asymptotically flat initial data to reasonable Einstein mother systems What about strong cosmic censorship? Well, this takes longer to understand why this is good But nonetheless it is widely considered to be good and that's why we want to conjecture So let me remind you. Well, you can ask in open-heimer Snyder. What happens to these incomplete? Observers, okay, so so happens first of all Related to what we just said that they they all have to pass into a black hole region but more over in finite time they reach this quote singular boundary are equal zero and Well, okay, of course, we all know that sort of the curvature blows up here Which is already a good start But actually something far more traumatic happens to observers. They're actually torn apart by infinite tidal deformations So why is this good? Well, it's certainly not good For the observer themselves But for the theory, it's good in that in a perverse way. This makes the classical theory deterministic Okay, the observers who don't go in the black hole they live forever theory predicts for them the observers who do again theory predicts exactly what happens to them from the point of view of Their sort of as long as they remain classical observers namely they are torn apart So in this sense General classical general relativity can be a closed theory no matter what the sort of What Harvey would call the UV physics sort of near r equals zero So this is exactly in some sense why One likes one would like sort of that property and What is the mathematician of this? So this is what strong cosmic censorship tries to capture So as we discussed last time it's sort of convenient to try to say this Without referring to observers per se, but just referring to the behavior of the metric itself Okay, and of course if you also had a lot of matter there, maybe you'd want to add an analogous statement for the matter but let me just Talk about the metric and my claim to you is that the way to formulate this is that the metric is inextendable as something okay, and One comparison I wanted to make is well the statement that the curvature blows up Okay already tells you that the metric is inextendable. Let's say as a C2 metric and I'll get back to this in just a second but actually the Sort of calculation you can do Concerning such observers This is related to the fact that the metric is inextendable as a continuous matter now that that actually is harder to prove than the Than just the statement about Some particular observers and well I discussed that a bit last time Let me make one remark that actually was was pointed out to me by a member of the audience Of course So the statement that the curvature blows up here You can you can capture by saying that let's say the metric is inextendable as a C2 metric, but actually that In in the Schwarzschild in the pure Schwarzschild case That's a that's a reasonable statement to make but in Oppenheimer-Sneider That's sort of a trivial statement because the metric is not C2 in Oppenheimer-Sneider On the boundary of the star and actually in general this points this this sort of Highlights an important point, which is exactly why talking about point-wise curvature Blow up is basically irrelevant in Discussing singularities because Actually in classical physics we routinely allow data Which is sort of weakly singular in some sense as long as we have well-posed in that class of data and Good example a classical example that we know in love in in pure vacuum general relativity is Penrose's impulsive gravitational waves Okay, where you know you you allow You have in fact delta function singularities in curvature. Okay, and these propagate and This is this is not considered to be a singular boundary of space-time. No, these are this is your solution and in fact It's a recent series of papers by Jonathan Luke and Igor Rudnjanski where they prove that you have a well-posed in a statement in that class So in a class of initial data that includes Those singularities and worse so if you're going to allow those already in initial data Then certainly curvature blowing up Is no other sign of sort of terminal singularity? okay, so This sort of tells you that you know whatever you want to write here. Okay, it should be considerably stronger than c2 Okay, and so see c0 would be you know the ideal if you want formulation of this conjecture and it certainly Suggested by by openheimer's night So we'll get back to this issue you should think that there's a there's a different version of strong cosmic censorship for whatever you Want to put here? Okay all right So there's one other point that maybe I'll add so So exactly because there's a load of confusion on the formulation of strong cosmic Sunship, which is in part historical So there's another property of this singular boundary Which is manifest from this picture whatever this picture is supposed to mean and we'll get back to that later on today Of course, and that is that this boundary is space-like Okay, so the the sort of the the boundary the boundary of space-time, which is associated to incomplete geodesics is space-like so Because this statement is often confused with strong cosmic Sunship Let me give it an independent existence and and formulate yet a third conjecture which I'll call the the space-like Singularity conjecture so So this conjecture would say something like the following so again for generic Sholdada so let me let me say it like this all incomplete Observer so this just means whatever time like or no geodesics Okay, and I'll write this in quote Meet a space-like singular so the reason I put this in quote is that it's not actually clear how exactly you would make sort of rigorous sense of this because It's difficult in general to define a priori boundaries of space-time Okay, for which, you know, this is sort of meaningful both This and also what it means to meet meet the singular okay, but at least that some vague sense You you can imagine What this means now? Why is this so I claimed it this has some connection to this and The connection you can think of it as follows that And we'll actually this will become more clear later in the lecture if you sort of don't follow exactly what I say But there is a sense that if if the boundary of space-time is as I've drawn it Okay, then this part of the boundary has to be singular at least in some sense and the reason is if if the boundary were not singular here Okay, then there is no reason why I could not continue space-time a little further this Still would be globally hyperbolic Okay, so this continued space-time should have been the maximum Cauchy development So somehow you would think provided that you could Formulate this okay that this would imply Some version of this but of course not necessarily with C0 here with something here okay, so Okay, I I'm not going to Try to Say more about that because sort of spoiler alert Both this statement and and and this statement are false So anyway, if you want you can still try to relate them Okay, so No, so I'll address I'll address this immediately in fact But it is a good point to re-emphasize and we'll see this immediately Now but these names are traditional and again There is a there's a reason for the names which maybe will be clear But in principle that they are completely different types of statements and in fact in in in more PDE language those of you who study PDE Then you should think of this as a as a statement of global existence well What's left of global existence after the singularity or the incompleteness theorem? All right, whereas you can you can think of this statement as a statement of global uniqueness Okay, so that's really so this is the statement that some observers they they live forever And okay, they don't see anything bad in finite time This is the statement that there's only one sort of space time that can possibly be associated to initial data and again morally at least if this is true then sort of Classical physics really ends locally at its boundary. Okay. All right, so Okay, but I think it's it's very important already before the sort of Before today's lecture begins. I warned you Beginning of last time that I will fall behind schedule. So today's lecture has not begun yet. We are still in yesterday's lecture So before talking, you know about Spirically symmetric space times in more detail where we can really make make sense of these diagrams And I think it's a very nice introduction to them Let me already tell you in what ways how could these properties fail whether different ways in which these properties could have failed So that you get some feeling for for for what these statements are meaning So let me begin with the sort of the first and maybe most well-known possible failure so you could imagine that you have great initial data, you know asymptotically flat with one end. I don't again I'm Purposely looking at Oppenheimer-Sneider as opposed to two-ended Schwarzschild and Kerr and Reissner-Northern They will those solutions I'll refer to them later just so that you don't think that this is this is about two-ended This is about real, you know astrophysical gravitational collapse in principle. Okay, so You begin with great dots like that. So there's a little bit of null infinity coming in like this The evolution is proceeding all fine and suddenly There's some sort of singular behavior at some point whatever that means Okay, and well since there's some singular behavior Cauchy evolution at best could live up to such a Null count so suppose that's exactly what happens so suppose this is the The Penrose diagram of your Cauchy development of of initial data, okay and and suppose that Sort of this null cone. Okay has the following two properties. So first of all, you know outside of this point Okay, this is completely smooth You can certainly imagine this this happens actually in all sorts of nonlinear field theories you have a First singularity, okay, but you know up to its future null cone the solution is Up to and including it's completely smooth And moreover you can imagine that this night light cone goes all the way to future null infinity in finite Time as measured at future null infinity. So in finite what you you should call bondy time Okay, so finite bondy time So this the boundary cutting future null infinity at finite bondy time is exactly The statement that future null infinity is not Complete okay in the middle of Sort of the LIGO run. They're all excited. They're going to announce some new discovery and all of a sudden well That's the end of the Cauchy development. Okay, because all of a sudden, okay, you are receiving Signals from very near here and okay. Well once once you're here one doesn't know how to evolve so So first of all this as I've just explained, okay, it means that We cosmic censorship is false or I okay. I am applying the usual Misuse of language which is fine as long as Namely, okay weak cosmic censorship is the statement that for generic initial data something holds So when I say weak cosmic censorship is false I'm just saying that the the predicate of that statement is False for this for such an example. Okay, so I will use that Abuse of language because it's obviously convenient. So weak cosmic censorship is false on account of this Okay, but so is strong cosmic censorship and Any any version of strong cosmic censorship is false. Okay Because well, I told you that the metric is smooth up up until this Sort of null cone so okay, I can just Extend space time here and in fact, you know, I can extend space time solving whatever equations of motion I think I was solving over here But moreover I can extend space time in an infinity of different ways and that's Why we want strong cosmic censorship to be true because you know, I have non uniqueness. Okay, so So this is if you want the the classic picture of a of a so-called naked singularity Okay, and you really should think that this this classic picture violates both Weak and sort of any version of strong cosmic censorship But you could also certainly imagine The following picture So again, you sort of start from your Great initial data regular nothing wrong with it. Okay Here's null infinity Bondi time is running over there all of a sudden there's a there's a sort of first singularity Okay And but now if you look at the sort of the future null cone from that sort of singularity That's also singular but you can imagine that moreover it still cuts off null infinity at Finite bondi time, okay, so So in such an example, okay, well weak weak cosmic censorship would be false Okay, but depending on how singular this is your your favorite version of strong cosmic censorship would be true Okay, so So they they they really are saying Different things okay. All right, so that's that sort of Important to keep in mind that in principle. These are really different statements, so let me give you another Penrose diagram to sort of Ponder so again, here's the data That is perfectly regular Now infinity bondi time is sort of Running along so you can imagine that okay again you hit the sort of first singularity but you you can imagine the following that the This first singularity is already behind an event horizon and Future null infinity is complete So no problem for weak cosmic censorship, but you could imagine that coming out of this first singularity. There is a there is a light cone Okay, and you can imagine that the solution is completely smooth on the boundary of this light cone, okay And then maybe okay, maybe there is some I don't know r equals zero like singularity here Maybe let's picture this let me write this but the metric here is smooth so Right, maybe I should say also Sort of funny since I mentioned the space like singularity conjecture the the this is a So The the space like singularity conjecture is false here Here strong cosmic censorship is true, but the space like singularity conjecture is false Okay And now let's let's let's think here. So weak cosmic censorship is true now Okay, but but strong cosmic censorship is false. I can I Can extend Okay, so any version if this is smooth any version is false Okay, and of course the the space like singularity conjecture is also false Because okay, maybe this part of the boundary is a space like but but I have this part of the boundary. That's not space So So it turns out that there's something there's something else that can happen So if you want this is this example is already a reason why historically In in in saying weak versus strong Historically people had this in mind So this is something let's say Which is not a counter example for weak cosmic censorship, but would be a counter example for strong But it's not the only possibility So let me give you something more sinister that you can have So again initial data is impeccable sort of Whatever if you think of this as being strictly symmetric, this is the center of symmetry Well, no problem with weak cosmic censorship. So whatever happens happens inside the black hole, which is beyond the Horizon and null infinity complete. I Always have to tell you that null infinity is complete And you can imagine that actually sort of near the center Maybe there's no problem though sort of the singular boundary is like an openheimer slider. I have this picture So maybe I have a sort of a null Part of the boundary that comes out of this point here, which is smooth So maybe this is again smooth so again, so let's just to to Say explicitly, why is this? Boundary of space time. Why can I not so say I can extend the space time? Why is not this part of my what I'm calling the space time? Okay by causality Okay, just for the same reason that this wasn't okay. This is not in the Koshy development of initial data But there's sort of a funny difference and this is very important to Always keep in mind. All right in this case You sort of know why there is this piece of the boundary. There's a first singularity Okay, and this is sort of a cone coming from a first singularity And what's very funny about this picture is that? Well in the Penrose diagram it looks exactly the same, but this isn't a singular point in any sense of the word So maybe I should say explicitly so in In all of these cases these boundaries are known as Koshy horizons This is a Koshy horizon. This is not a Koshy horizon in At least in the class that this is singular So it is the boundary of space time in a sufficiently regular extension Okay, and the the reason that again this is consistent with this being the maximum Koshy development is these extensions fail to be globally so So the point is that Koshy horizons they they can sort of be generated by Sort of first singular points like we see here But they can also not be generated by by first singular points like here And let me in some sense give you the the most sinister example of them all so the most sinister example is the following So the most sinister example is perfectly regular initial data Perfectly complete future null infinity Okay, so this is complete But the Koshy development ends in a Koshy horizon Okay, which is entirely coming from this point and there's no other boundary in this case Sort of if the metric is completely smooth here, okay, then you can extend the metric smoothly Still satisfying the equations of motion completely non uniquely so that every single incomplete observer in the original space time Passes safely to the extension so every observer that was incomplete in the original space time can pass to the extension so What is this telling so first of all this is why? One should never utter the words Penrose singularity theorem. Okay, because this is exactly Completely in the domain of that theorem. Okay, this is a this is a trapped surface. Okay, this is the maximum Koshy development Of a space time. Okay, you know Satisfying this is the quintessential thing that you would apply Penrose's incompleteness theorem to and Penrose's incompleteness theorem successfully tells you that it is Geodesically incomplete. Okay, but it is not geodesically incomplete because any Observer is encountering a singularity is geodesically incomplete because it can be extended and it can be extended smoothly Satisfying the equations of motion so that all observers who were incomplete pass into the extension Okay, the problem with the extensions is just the extensions cannot be globally hyperbolic. Okay So the extensions are not uniquely determined by initial doubt. Okay, so So fundamentally that that theorem. Okay is Not distinguishing between any of these behaviors and in some sense You really should think that morally the proof is connected with this behavior and not with any of the other It really is not a singularity theorem. Okay, so Now of course already and I'll talk about it later. This is not some exotic thing This is just the one-ended version of the curse solution. Okay, so this phenomena happens in the curse solution and in its spherically symmetric sort of Cousin rise or Nordstrom. Okay, so in that case you have two-ended initial data and The solution looks like this And this is a kosher horizon and again if you want you can you can apply Penrose's incompleteness theorem to this and it's telling you that this is geodesically incomplete which it is and That's that's true. All these geodesics are incomplete, but you can extend in in a Infinity of different ways, none of which is better than any other Such that all these incomplete observers gets by. Okay, so this This is how Weak and strong cosmic censorship can fail. Okay, so the the conjectures are trying to say Okay, that all those things do not happen generically. Now One thing should be clear immediately is that you do need the caveat Generic and maybe I should say this Very explicitly at the beginning you certainly need this And this was clear from the very beginning for strong cosmic censorship slash space-like singularity conjecture because of course the Kerr solution was known Okay, beforehand and okay if you're willing to Allow two-ended data or if you put something in here, which you can do to create something like this then This is a counter example. So immediately You know the best You could hope for was was generic when weak cosmic censorship was originally Conjectured it wasn't clear that you needed Genericity But it turns out and this this is something I'll talk about later it turns out that that you do now Maybe I should say though very clearly Already one important thing So always when I say initial data it is initial data for what you know for so you should think in all of these Conjectures it should always be initial data Let me let me write it because it's so important for for a reasonable Einstein matter System so what what is a reasonable Einstein matter system? So So well The vacuum equations they should certainly be reasonable if that's not reasonable then nothing is reason okay So certainly these conjectures should hold for vacuum. They should hold for electro vacuum But once you start coupling with mother you might have problems so One of the big ironies actually of Oppenheimer-Sneider is the following so remember Oppenheimer-Sneider was a Solution of the Einstein dust system so Einstein perfect fluid where the perfect fluid is Pressureless, okay, so So of course this played a very important role in Illucidation of the notion of a black hole in our picture of gravitational collapse and if you remember What was true about initial data for for the ancient for Oppenheimer-Sneider? So the density initially was constant. Okay, so it was constant Inside what we think of as the star and vanishes outside This was this was the model So it turns out that within spherical symmetry actually if you perturb a little bit The initial state keeping spherical symmetry, but not having constant density You immediately get something like this generically and If you perturb a lot you actually get this so So at least within spherical symmetry weak cosmic censorship is actually false For this model okay and strong cosmic so everything is false for this model but But that it's not It is not correct to interpret that as a failure of sort of the spirit of these conjectures because somehow this mother model, okay is a very Sort of a special type of mother model because it does not take into account pressure Okay, you can think of this as an idealization of a perfect fluid with a reasonable equation of state okay, and Actually sort of what happens in this case is that the the density blows up here And when the density blows up you might think that okay pressure is actually important so even if your equation of state was such that okay, you're sort of You're ignoring pressure and you're modeling by dust when you get near here pressures so So in any case What what this is sort of telling you is that mother models that sort of on their own Give rise to sort of very singular behavior. Okay are not good mother models to try to address these questions in Okay, and so, you know even some of the you know most famous mother models in sort of the story of gravitational collapse are actually very pathological in reality and you know are not You know in the class of reasonable Einstein mother models. Okay, so this is something to to to keep in mind Sort of what what you should think of as sort of a safe mother model a mother model that you certainly would want this These conjectures to apply to is a mother model where the equations of motions themselves are linear Okay, that that should certainly be safe Okay, because those linear equations do not form singularities of their own accord Okay, so Einstein scalar field is one such example Einstein Maxwell Another nice example of that type is Einstein Vlasov. So Einstein coupled to collisionless Boltzmann and all these sort of are nice models. They even have sort of some Sort of physical interpretation in some sense in Realistic problems, but Conceptually, they're sort of very useful mother models to keep in mind. All right In particular one one should remember if one wants to consider these problems Restricted to some symmetry class for instance to spherical symmetry. You have to add in mother by Birkhoff's theory So sort of Very often in the context of the study of these problems in spherical symmetry. Okay, the choice of the mother is not Because we believe in that particular mother model But it is you can think of it as a sort of a model problem Okay to understand even the vacuum case without symmetry and if that's how you're Thinking of your mother then you certainly don't want the mother to have pathologies of its own Which are very different from the vacuum And that's sort of what actually happens with with a failure of these statements for Einstein dust all right, so With all that said, let's begin lecture two So lecture two is basically I want to make sense of what what these pictures mean and then Lecture three. I'll tell you What sort of what we know in spherical symmetry and lecture four well I'm very optimistic What what what we know definitively not in So All right, so So the the sort of title of this lecture two is a spherically symmetric dynamical space times And panoramas and this is a as I said, it's a great world for understanding all sorts of issues connected to gravity There's so much of I mean everything that we drew can be understood in this world. Okay so So first of all note the word dynamical Okay, I will always think of space time as being the crochet evolution of initial data because that's So general relativity as a theory after all is that it's the thing that associates, you know this Unique solution to the initial problem. Okay, so And moreover, I'll I'll I'll always be living in the asymptotically flat world Although as we sort of discussed last time and we saw In the last lecture of toby, it is also of interest to consider other asymptotics like asymptotically ads Sort of space time then of course, it's no longer a pure initial value problem. It's an initial boundary value problem But you can you can certainly discuss that world in much the same language that I will talk about Okay, but I will only look at the asymptotically flat case So, uh, I first have to tell you what asymptotically flat historically symmetric initial data Look like Okay, so, um So initial data. So this is a, um A three manifold and if you want the the the manifold topologically, okay Can either be r3 Or s2 cross r those are the only two cases Okay, this is the so-called one-ended case And this is the the so-called two-ended case And uh, I can write the metric This is the initial metric. This is initial data. Okay, I can write the metric if you want I can think of So it's sort of funny R3 you can think of it again as the sort of warped product of S2 cross 0 infinity. Okay, so I don't know how to Write that. Okay, so some warped warped product. Okay And uh, I've already written it backwards. So let me write r first Okay So I'm going to write but write the metric like this Um, so normally you might think that the the coordinate here should be called r But I'm going to call it x and you'll see exactly So I'm going to write the the initial data metric is some some function h Of x dx squared Okay, plus um And this is why I wrote this as x because I want this to be r. Okay And here I'm going to write the Standard As spherical metric right so So what you should think is that in in in the one-ended case Okay, this is the one-ended case Okay So when you write the metric like this you see what this r means geometrically r is the sort of Modular factors of 4 pi it is the the the area Okay of these spheres. So I want r to be the area of those spheres. Okay, so It could be in this case here that I can also use r as a coordinate. Okay, but I might not be able to okay because No one is telling you that r should be monotonic. Okay, and in fact in the two-ended case Okay r cannot be monotonic So it's clear that I I I could I could never use r as a coordinate globally on initial data Because of course r goes to infinity here And r goes to infinity There So here r is equal to zero And r goes to infinity r could be monotonic, but it might not be Generally, it won't be Okay So so that that is one part of initial data. Of course, I also need to prescribe the second fundamental form Okay, and well In general in in spherical symmetry you you better be solving An Einstein mother system because otherwise the only solution is Schwarzschild. Okay, so there'll be a bunch of fields Which will also need their initial data, but I'm not going to Refer to them at this point. I'm not going to specify The sort of mother model I'm going to make general statements about the geometry. Okay. All right, so So So what I'm going to suppose is simply that Whatever the mother fields are okay, they They do Determine a well-posed So this is the notion that Harvey talked about In the morning a well-posed system of equations. So just like for the pure volume we can talk about Maximum Cauchy development, which will be a globally hyperbolic space time Satisfying the equations of motion and admitting this as data. Okay, so Let me write that so I'll Suppose m comma g comma bunch of mother fields Okay Is a globally hyperbolic Space time arising initial data From sort of spherically symmetric initial data, so Initial data of Type one or two, let's call this type one. Let's call this type two. So So I want to draw A picture of of m or some representation of m. So here is my claim And this will explain the mystery of these Diagrams so so point one Is Then M is spherically symmetrical So geometrically that means that so three acts by a geometry. No, this is a Consequence I Let me emphasize this fact this follows from well-posedness if you want If your initial data Are Have some continuous symmetry Okay, then the solution the Cauchy development of initial data will inherit that symmetry This is a corollary. You can think of it as a corollary of uniqueness Okay, so that's a good exercise. All right, so Remember assumptions can only be made at the level of initial data. Yes But covering the whole thing is not the same as being a coordinate. It is a to be a coordinate it has to be so in the in in one dimensional so to You know relative to the the actual coordinates you understand some Jacobian has to be non vanishing so in one dimension It just means that the Right, so so if if r as a function of x Okay It's derivative vanishes at some point then that tells you that arc r is not a chord All right, um I mean it could be worse even that's to say, uh, you know R can even be constant, you know in an interval like that. There's no you know That's also allowed. It actually happens in What's called nary I solution Anyway, um, okay So then m is working symmetric. Um, and more over and can be covered By global double null coordinates u and v i.e You can write the metric globally. Okay as G equals some function, which I'll call minus omega squared of u comma v d u d v plus Some other function r squared Of u comma v times the sort of standard metric on the sphere. So if you want, uh, u v Theta phi are global coordinates. Of course, okay global modulo the trivial. Okay fact that You lose, uh, right A great circle, uh, half of a great circle on on on the sphere. Okay um, so, um So This is actually a good, uh This is a very good exercise already. Okay, this one is using something. I mean first you have to infer This the spherical symmetry is is preserved and then why are these coordinates global? Well, this this has to do with the fact that m Is again by fiat globally hyperbolic. So it is globally hyperbolic And I've told you something about the initial data. It sort of looks like this or this So already this is a good exercise. But some of the things I'll say in just a little bit of time will maybe Throw some light on how how you would already prove this statement. Um So let me make a A trivial Amplification of the statement Um, so null coordinates are manifestly non unique. Okay In particular, I can always rescale, uh, u I can always, uh Consider a new u tilde to be an arbitrary monotonic in order to be a strictly monotonic function of u and and uh, sort of, uh, also, uh V so if I have null coordinates for for any choice of of strictly Monotonic, you know functions or strictly I should say, yeah, such that f prime Let's say without loss of generality is is greater than zero and g prime is greater than zero then I have a new null coordinates. Okay Um, so in particular by your your your favorite dif eomorphism of, uh, you know R to you know the interval minus one one. Okay, you can always, uh, assume that these coordinates are bounded That's to say their range is bounded Okay, so let me write this here already by bounded so, um So now I can think of the the bounded null coordinates as defining a bounded map um So I can think of u comma v Okay, as defining a bounded map of the manifold m Okay to r2 And I want to think of r2 if you want as r1 plus one Okay, and I want to make the following identification. So, uh, think of, um ambient coordinates, uh on, uh r1 plus one as t and x Okay, so let my identification be this so, uh so if if you want the the The image of m by the above map, okay, so I'll denote it Script p. So it's a subset. It's a bounded subset of r1 plus one. Okay Uh, I I call this the Penrose diagram of of m. Okay, so this is the Penrose diagram. It's just the the range of a bounded, uh Uh of a bounded set of double null coordinates Okay All right So the image Is called the the the Penrose diagram. Okay, so, um Were I a a mathematician Maybe I would say a Penrose diagram Because of course, okay There are mentors lots of choices of double null coordinates. So this is not as I defined it a unique object But it turns out that somehow it's it's it's Everything I'm going to say about these objects sort of are are canonical. Okay, so you can If you're that type of a person you can you can sort out that On your own time. Okay So, uh, that's the Penrose diagram. So let's uh, um, so let's um draw so first of all, um I'm uh unapologetically progressive So, uh, as a result, uh, actually I'm only going to draw The the future goshi development Okay, because I forget about the past I I will look Only towards the future. Okay, so, um, so from now on m comma g will be the the future Okay, we don't look back Okay, so, um, so what does what does it look like? Um, so maybe I'll erase this, uh, but I won't erase that because that may be useful so let's distinguish, uh, between types one and and type two actually, uh, maybe It's better to start with type two because type two is is actually You have to think less about it. So, um So in particular the initial data itself, okay as, uh, uh Sort of subset. So this is r two now thought us as r one plus one Okay, so you should think, you know, my ambient coordinates would be, you know t D by dt is like this D by dx is like this but in view of my identification This means that d by du is like this And d by dv is like this Okay So, um, so I want to draw P. Okay, so this is if you want What will be a past boundary of P? Okay, so what what does P look like? So here's a little exercise Okay, whatever P is Okay thought of as a subset of r one plus one It is globally hyperbolic. Okay, so it's a globally hyperbolic subset Of minkowski space in one plus one dimensions. Okay, so I claim that okay All right, I'll explain this picture in just a second, but so this is P. Okay, so this is a globally hyperbolic So this is the two-ended case so this is Type two if you want globally hyperbolic As a subset of r one plus one Um And I'll explain Um Somehow what uh, well, maybe I'll say Immediately what what that means that means that this P. Okay, you can write it as a union Okay of uh closed Past null cones, okay intersect the the future of this As a subset of again this ambient minkowski space So it is it is a union of closed past cones So in particular, this is telling you that Given any point in P. Okay, if you follow this direction backwards Okay, you will you will reach initial data and if you follow this direction backwards you will reach initial data Okay All right Uh, so we'll we'll we'll come back to sort of the Okay, but the implications this has sort of the boundary of this et cetera in just a second Let me immediately, uh, so let me just so that this picture gets some content. Okay So um, so with this way of writing the metric, of course, this r Restricted to initial data core coincides with this r. Okay So actually again by choosing the null coordinates if you want you can make this exactly horizontal. So that's another exercise So what about type one? So remember in type one Okay, which you really should think is the physical case okay in In in the real world, maybe not in some speculations in high energy physics, but in sort of astrophysics certainly You know, we you know the the initial Sort of the initial data does not have two asymptotically flat ends. It has one. Okay, so so So somehow in in in in the one-ended case This this p has a piece of boundary that is coming up from here On which r equals zero Okay, so these are actually points of the spacetime Okay And what what what what does this correspond to well again if you're thinking geometrically Spherical symmetry means I have this s o 3 action acting by isometry on the manifold and The points here are actually points upstairs Okay, they are not spheres. They are points Okay, so they are the fixed fixed points of the the action Okay, um, so it turns out that the the set of such points upstairs is actually a time like judy's So so you get this sort of boundary and you should think this boundary is part of the spacetime Okay, and we already saw this in the case of openheimer snyder. Okay, we saw this already. Okay, so In view of this note that okay, so so so p then looks like something a priori it could look like this Okay, of course, this is cannot be globally hyperbolic because well here here is a sort of Inextendable Null curve that never gets to Sort of initial data. Okay, so the statement is that if if you take all points in p Okay, if you follow null curves in this direction, you'll hit either r equals zero this so this I'll I'll denote this sort of part of the boundary By couple rama. Okay, so you'll either hit rama or You'll hit So let's call this sigma. Okay, so sigma is the Initial data the the initial manifold. Okay So I'll use sigma both for maybe this is already abusive notation For for this and for its projection, okay So So the property is if I go this direction, I either hit the rama or or sigma and if I go this direction I always hit so So everything we did so far just follows from the fact that these are Spherically symmetric Globally hyperbolic space times which admit the initial data of that form. Okay, that's Everything so far as we have just used that and everything in some sense completely elementary. Okay, so So so this is What a penrose diagram is in in spherical symmetry. Okay, it's something very Concrete and specific now. What what is it good for? so Let many of you sort of know and in some sense we've already Obligately referred to y but I should sort of spell it out. So it's good for for two reasons one We can immediately infer certain causal relations in p so in particular if this is Sort of a point here, of course a point is a sphere Upstairs unless it's here in which case it's really a point Okay, then I can look at the the future of this point in one plus one dimensional minkowski space That's this region here And the claim is again, this is an exercise in just lorenzian geometry given the form of the metric that This sort of uv coordinate range Corresponds exactly to the future of this sphere in this metric Okay, so I can I can immediately see the future And similarly the the past Of spheres, okay, and of course the the the future of any point actual point in the spacetime that lives on this sphere Is contained in the future of the sphere, right? so somehow I can I can see immediately that the sort of You know if I have a point an actual point here and a point here that you know the past of this point Is is contained in the past of this sphere the future of You know a point projecting to that is contained in the in the future of this sphere So these don't intersect so I I know something about causality upstairs, okay Now be careful. You can't read off everything. So you don't know exactly, you know if I take the the The future of the point on this sphere Okay And then I re-project it to this picture. Okay, I don't know exactly what it looks like Okay, maybe it looks sort of like like this. Okay, but I know it's contained in here So With that caveat, you know, you can say lots of non-trivial things about the the causal structure But the story gets bother because you now have a boundary You have a boundary trivially simply because this is sitting as a as a bounded subset of R1 plus 1. So you have the boundary of p in The ambient r1 plus 1. Okay, and of course the boundary. Okay, let's maybe first say the two-ended case Okay, because I'm progressive. Okay, my initial data Is part of the boundary, but I also have this Okay, and moreover because this is all sitting in r1 plus 1 I can apply causal relations Using points at the boundary Okay, so I can look at this point at the boundary and say oh what is the past of this point intersect p And I get something so anyway That's why panels diagrams are so Useful you get for free this boundary now. Let me be completely explicit about a very important Point before I talk more about the boundary of p The boundary of p is really a boundary of p It is not a boundary of spacetime I am only Going to be discussing the boundary of the panels diagram Okay, now you might try to use that boundary to define some boundary of spacetime upstairs Okay, but that in general is a non-trivial Okay This boundary is completely trivial It's really completely trivial, but that you should think of it as a good thing Okay, that means that we're not making any choices. We're not this is really a completely canonical thing Okay, but remember this is a boundary downstairs and downstairs only Have to remember that all right. So what what does this boundary look like in general? What can we say about the boundary of p And again, it's easiest to first look at case two so so Let me okay do case two first and I claim that so everything I'm saying is elementary causality of One plus one dimensional minkowski space Okay, I am just using the fact that p whatever it is is globally hyperbolic as a subset of One plus one dimensional minkowski space with this as a cushy hyper surface. So I'm using nothing else Okay, so what can the boundary look like? Okay. Well, of course, okay. This is part of the boundary the initial The initial data itself Okay, with these sort of two Endpoints of initial data which correspond to the asymptotic flat ends. This is the two-ended case so one thing you might have Is Boundary components that come out of the initial data like this Okay, because obviously if you had any points here This could not be a cushy hyper surface, right? So you could have boundary components like this Mind you, you might not have that. So these could be empty So you could have these um Now, of course, these might actually close off the whole space time. That's one possibility So if they don't then you have to have at least one First singularity. So what I'll call a first singularity First singularity First does not mean there's only one by the way. It's just you'll see what first singularity means So first singularity means it's a point on the boundary Okay, such that if I erect this Characteristic rectangle in minkov in one plus one dimensional minkowski space the whole rectangle Is in space time is in p except for this point All right, so note if these close up Then there's no first singularity. Okay all points on the boundary, you know part of the you know This part of the rectangle would not be In the space time, right if I take this point here and erect that rectangle, right? This whole edge is not in the space okay, so So I know if if if these don't close up that I have at least one first singularity, okay, maybe I have several Okay, maybe the whole boundary is first singularities Okay, but maybe coming from first singularity. I have two null pieces so it turns out that What you should think is The boundary, okay So the boundary looks as follows, okay, it's This possibly empty Union this possibly empty. Okay union Some number of first singularities possibly uncountable possibly continuum Okay, but possibly to each first singularity. There is one or two Null components that emanate from it. Okay Now again By the way singularity don't read anything into it, okay Although, you know, if this is a true Cauchy development, there will have to be something singular about such points, but at this You know What I've said is just using global hyperbolist nothing else Okay, so Let me give you a an example immediately or two examples that we know and love. Okay, so Um, yeah Yeah, yeah so example so, uh So Schwarzschild, okay Um Well, maybe I'll say rice or north from first because it's sort of simpler So rice or north from from the point of view of what I've just said The boundary looks like this Okay That is rice or north And so it's really the case that you have no first singularities Okay, um Schwarzschild, okay, you have this okay, and then uh everything else Okay is is a first singularity So everything else all the other points Are first singularities. So if you draw the characteristic rectangle here Then the the rectangle is completely within the spacetime Except for those points. All right, but you could imagine uh a priori Okay, a very complicated situation where maybe there's some Um Fractal set of uh first singularities Okay, which some of which have you know these null components, but you know, there's it's really a fractal set something like that Okay, that's completely Within the realm of possibility. Okay all right, um So let me very quickly tell you, um, how is this modified in? in case one So in case one of course, uh, you have, um Also this boundary Rama Okay, which is secretly it's it's it's not a boundary Upstairs of spacetime. It's uh, it's the fixed points of the s o 3 action. Okay, so So of course, um, uh, so what could happen? Well again, of course in in general you might have a a null Piece that comes from here. Uh, and actually this null piece could could you know meet this boundary? Okay, so let me write that there Uh Does that ever happen? Minkowski space. Okay, so that's that's not so exotic After all, okay, so that could happen But um in general, okay, there could be a null piece coming from here Okay, and then everything else Is either a first singularity Or a null piece coming from a first singularity. So everything else is a null piece or Is a first singularity or a null piece coming from first singularity. Okay, so I so openheimer snider if you want Right Looks like this Okay, so all these were first singularities Okay, and this was the the null piece. Okay, but I also drew some examples earlier today where Where where this was a where this was a null piece Okay, so So let me just uh make in the last two minutes It's not so bad. We're well into lecture two. So I think it will be fine and but let me let me Make some So Toby liked to have some philosophical comments or ideological quotes So let me make sort of some ideological comments um In view of the fact that that that schforzschild and and reiser nordstrom are On the board okay, um, so In textbooks one reads that you know This picture here. Okay, maybe also with a with a past but okay as I said I am progressive this picture here is maximally extended schforzschild is the maximal analytic extension of schforzschild as if the procedure of sort of Coming up with spacetimes was having spacetime in some patch and analytically extending the metric. Okay, that's not Sort of how spacetimes arise spacetimes arise by solving the initial value problem spacetimes arise dynamically so The reason that this deserves to be called schforzschild is if you want complete initial data Which are spherically symmetric then you have to draw this And then this object is the maximal kosche development so schwarzschild the sort of Cross-call extension of schwarzschild which actually appears in a much earlier paper of sing Anyway, that object you should think of it. Not as Maximally analytically extended something or other it is the maximal kosche development of complete spherically symmetric initial data So that's it. We agree on what the object is, but you should think about it in those terms Okay What about riser northstrom and kerr? Well, it's the same Okay, at the end of the day what deserves to be called the riser northstrom metric or the kerr metric Is the maximal kosche development of riser northstrom or kerr initial data And uh as and i'll interpret all these things later on but as I drew informally earlier So I can restore them now We know that I can further decompose this boundary in in this sense And this part of the boundary is a kosche horizon and you can extend beyond And okay. Well in in textbooks you often see a particular Extension in the kerr in the riser north some case that has singularities that has whatever that but There is nothing unique about that extension From the point of view of kosche problem um In particular one one thing that one often sees is in in a particular extension you have some So-called time-like singularities here and people often think that oh, this is the kosche horizon Of a time-like singularity just like you know, if you thought of asymptotically flat gravitational collapse with one end and sometimes in Anti-diluvian literature you see a time-like singularity here and this is a kosche horizon of this sort of time This is fantasy You can you can never talk dynamically about time-like singularities Exactly because there are time like okay, so So in any case one one really has to learn to talk about all these questions without ever Sort of thinking drawing any particular You know extension here and and well certainly not You know giving any weight to any particular property of you know extensions defined by Real annoticity or something like that this this that's fantasy So, uh, so it's really the case that uh one one should call this if you want the the riser nor some solution or you know the curse solution and well, okay, it's sort of The extension of this you get from annoticity you can one should call it something else I mean it is an entertaining thing to look at but it it has nothing to do with with dynamics in in in general All right, so we'll continue next time